> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# Legendre Base-Space Program Phase 13
## Square-Anchored $P_2$ Constant Ledger
### The Rough-Lower Gate, Bilinear Semiprime Upper Bounds and a Parity-Signed Reformulation

**Version:** v0.1  
**Date:** 2026-10-02  
**Author:** Neo.K × AI collaboration  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Research line:** Legendre Base-Space Prime-Fitting Program  
**Prerequisites:** Phases 1–12; SRBRD-13–20 provider audit  
**Status:** Phase 13 constant-ledger audit; main-term crossing PASS, current black-box provider crossing FAIL; Legendre's conjecture remains unproved

---

# 0. This Phase Does Only One Thing

Phase 12 established the exact $P_2$ boundary:

$$
\boxed{
P_\square(n)
=
S_3(n)-E_2(n),
}
$$

where:

- $P_\square(n)$: the prime count in the square cell;
- $S_3(n)$: the cube-root RootRough survivor count;
- $E_2(n)$: the semiprime count among them.

Phase 13 adds no new ontological layer.

The only question:

> After normalizing both sides entirely to $H/\log X$, can the constants of current analytic providers prove
>
> $$
> S_3(n)-E_2(n)>0?
> $$

---

# 1. The Standard Scale

Let:

$$
X=n^2,
$$

$$
H=2n\asymp X^{1/2},
$$

and:

$$
z=X^{1/3}.
$$

Ignoring integer rounding:

$$
\log z
=
\frac13\log X.
$$

---

# 2. Exact Boundary Identity

Phase 12 proved:

$$
\boxed{
S_3(n)
=
P_\square(n)+E_2(n).
}
$$

Thus:

$$
\boxed{
P_\square(n)
=
S_3(n)-E_2(n).
}
$$

This is the only ledger identity of Phase 13.

---

# 3. Ideal Prime Constant

The short-interval prime heuristic / PNT main scale:

$$
\boxed{
P_\square(n)
\sim
\frac H{\log X}.
}
$$

Thus the normalization is:

$$
\boxed{
C_P^*=1.
}
$$

---

# 4. Ideal Semiprime Constant

The semiprime least factor:

$$
p=X^t,
\qquad
\frac13<t\le\frac12.
$$

The switched cofactor interval has length:

$$
\frac Hp.
$$

The prime-density diagnostic gives:

$$
dE_2
\sim
\frac H{\log X}
\frac{dt}{t(1-t)}.
$$

Thus:

$$
C_{SP}^*
=
\int_{1/3}^{1/2}
\frac{dt}{t(1-t)}
=
\boxed{
\log2.
}
$$

Numerically:

$$
\boxed{
C_{SP}^*
\approx
0.69314718056.
}
$$

---

# 5. Ideal Rough Constant

Therefore:

$$
C_R^*
=
C_P^*+C_{SP}^*.
$$

That is:

$$
\boxed{
C_R^*
=
1+\log2
\approx
1.69314718056.
}
$$

This also equals:

$$
3\omega(3).
$$

---

# 6. Ideal Constant Ledger

Thus the ideal ledger is:

$$
\boxed{
(1+\log2)-\log2=1.
}
$$

This means:

> There is no constant deficit at the main-term level.

The actual difficulty lies entirely in pointwise error / provider loss.

---

# 7. Band Split

Split the semiprime least-factor exponent into:

## Low band

$$
\boxed{
\frac13
\le t\le
\frac25.
}
$$

Its ideal coefficient:

$$
C_A^*
=
\int_{1/3}^{2/5}
\frac{dt}{t(1-t)}
=
\boxed{
\log\frac43
}
$$

Thus:

$$
\boxed{
C_A^*
\approx
0.28768207245.
}
$$

## Middle / high / balanced aggregate

$$
\boxed{
\frac25
\le t\le
\frac12.
}
$$

Its ideal coefficient:

$$
C_H^*
=
\int_{2/5}^{1/2}
\frac{dt}{t(1-t)}
=
\boxed{
\log\frac32
}
$$

Thus:

$$
\boxed{
C_H^*
\approx
0.40546510811.
}
$$

Verify:

$$
\boxed{
C_A^*+C_H^*
=
\log2.
}
$$

---

# 8. Abstract Provider Multipliers

Suppose a rough lower provider gives:

$$
\boxed{
S_3(n)
\ge
C_R^-
\frac H{\log X}.
}
$$

Suppose upper providers for the low / high semiprime bands give:

$$
E_{2,A}(n)
\le
R_A
C_A^*
\frac H{\log X},
$$

$$
E_{2,H}(n)
\le
R_H
C_H^*
\frac H{\log X}.
$$

Then:

$$
\boxed{
P_\square(n)
\ge
\left[
C_R^-
-
R_A\log\frac43
-
R_H\log\frac32
\right]
\frac H{\log X}.
}
$$

The crossing gate is therefore:

$$
\boxed{
C_R^-
>
R_A\log\frac43
+
R_H\log\frac32.
}
$$

---

# 9. Common-Multiplier Threshold

If:

$$
R_A=R_H=R,
$$

we need only:

$$
R\log2<C_R^-.
$$

That is:

$$
\boxed{
R
<
\frac{C_R^-}{\log2}.
}
$$

Threshold table:

| rough lower constant $C_R^-$ | largest common semiprime multiplier $R$ allowed by $C_R^- > R\log2$ |
|---:|---:|
| 0.800000 | 1.154156 |
| 1.000000 | 1.442695 |
| 1.200000 | 1.731234 |
| 1.400000 | 2.019773 |
| 1.500000 | 2.164043 |
| 1.600000 | 2.308312 |
| 1.693147 | 2.442695 |

If the rough lower bound can approach the ideal:

$$
C_R^*=1+\log2,
$$

the semiprime upper bound can tolerate:

$$
\boxed{
R<2.44269504089.
}
$$

In other words:

> The semiprime provider need not approach the asymptotic constant $1$ for crossing to be possible.

---

# 10. Band-Specific Slack

If the high band has attained the ideal:

$$
R_H=1,
$$

the low band can still tolerate:

$$
R_A
<
\frac{
1+\log2-\log(3/2)
}{
\log(4/3)
}.
$$

That is:

$$
\boxed{
R_A<4.47605949678.
}
$$

Conversely, if the low band is ideal:

$$
R_A=1,
$$

then:

$$
\boxed{
R_H<3.46630346238.
}
$$

This again shows:

$$
\boxed{
\text{The semiprime upper constant is not the first bottleneck.}
}
$$

---

# 11. Provider A: A Square-Anchored Rough Lower Bound

What we actually need first is:

$$
\boxed{
S_3(n)
\ge
C_R^-
\frac H{\log X}
}
$$

holding at **every square anchor**.

Here:

$$
H=X^{1/2},
\qquad
z=X^{1/3}.
$$

---

# 12. Standard Linear-Sieve Gate

For a consecutive interval:

$$
(X,X+H],
$$

there is the local model:

$$
A_d
=
\frac Hd+O(1).
$$

The natural level:

$$
D\lesssim X^{1/2-o(1)}.
$$

Thus:

$$
s
=
\frac{\log D}{\log z}
=
\frac32+o(1).
$$

But the standard linear-sieve lower function satisfies:

$$
f(s)=0
\qquad
(s\le2).
$$

Therefore:

$$
\boxed{
C_R^-=0
}
$$

is the result this black-box provider can guarantee at this scale.

Thus:

$$
\boxed{
\text{standard linear lower sieve}
\text{ fails immediately at the first ledger entry: FAIL.}
}
$$

---

# 13. This Precedes the Semiprime Constant Problem

Even if the semiprime upper bound is perfect:

$$
R_A=R_H=1,
$$

if:

$$
C_R^-=0,
$$

we still have:

$$
0-\log2<0.
$$

Thus:

> In the current architecture, the first-priority missing theorem is not a more precise $E_2$ upper bound, but a square-anchored cube-root rough lower bound.

---

# 14. Generic Jacobsthal Is Still Insufficient

Being $z$-rough is equivalent to avoiding all prime divisors of the primorial:

$$
P(z)=\prod_{p\le z}p
$$

as specified above.

Generic arbitrary-translate existence is controlled by the Jacobsthal function.

The natural scale of an Iwaniec-type generic upper bound is about:

$$
O(z^2)
$$

up to logarithmic factors / a reformulation in terms of $\omega(P(z))$.

For:

$$
z=X^{1/3},
$$

this is:

$$
X^{2/3+o(1)},
$$

still greater than:

$$
H=X^{1/2}.
$$

Thus a generic Jacobsthal guarantee cannot give the pointwise rough survivor needed here.

---

# 15. Campbell's $P_3$ Is Not a Raw Rough Lower Bound Either

The all-square $P_3$ theorem of 2026 uses:

- Richert logarithmic weights;
- An explicit linear sieve;
- Finite verification.

It proves:

$$
\boxed{
\forall n,
\quad
(n^2,(n+1)^2)
\text{ contains }P_3.
}
$$

But:

$$
P_3\text{ existence}
$$

is not equivalent to:

$$
\boxed{
S_3(n)
\gg
\frac H{\log X}.
}
$$

Thus Campbell's theorem cannot fill Provider A directly.

---

# 16. The Correct New Target for Provider A

A pointwise lower bound with some:

$$
\boxed{
C_R^->0
}
$$

must be established.

The ideal target:

$$
\boxed{
C_R^-
=
1+\log2-o(1).
}
$$

But the ideal value is not necessarily needed for constant crossing.

It suffices to jointly satisfy, with the upper constants of Provider B:

$$
\boxed{
C_R^-
>
R_A\log\frac43
+
R_H\log\frac32.
}
$$

this crossing condition.

---

# 17. Why Must the Square Anchor Be Used?

Generic interval rough sieving sees only:

$$
X+k.
$$

Here we have:

$$
X=n^2
$$

and:

$$
n^2+k
\equiv0\pmod p
\iff
k\equiv-n^2\pmod p.
$$

Thus each local obstruction class carries the quadratic phase:

$$
\boxed{
-n^2\pmod p
}
$$

of this anchor.

Phase 4 proved that intersections of different prime obstructions retain the same square anchor:

$$
k\equiv-n^2\pmod d.
$$

Thus the only reasonable breakthrough point for Provider A is not fine-tuning generic sieve constants, but:

$$
\boxed{
\text{square-anchored discrepancy / phase cancellation}.
}
$$

---

# 18. Provider B: An Exact Bilinear Semiprime Upper Bound

The exact switching of Phase 12:

$$
E_2(n)
=
\sum_{z<p\le n}
\left[
\pi(U_p)-\pi(L_p)
\right].
$$

is equivalent to:

$$
\boxed{
E_2(n)
=
\#\{
(p,r)\in\mathbb P^2:
z<p\le r,
\quad
X<pr<X+H
\}.
}
$$

Thus Provider B is:

## Short Hyperbolic Prime-Convolution Upper Bound

---

# 19. The Transferable Parts of SRBRD Technology

SRBRD-18–20 audited:

- Maynard-II: the low Type-II factor range reaches $X^{2/5}$, but the original theorem fixes the residue $a$;
- Maynard-III: middle/high factor convolution has uniform residue estimates;
- Near $2/5$, there is a bridge proposition;
- Balanced convolution has a restricted-uniform result;
- Prime indicators satisfy the relevant coefficient class;
- Pascadi / Yang provide stronger structured-weight distribution technology.

Thus, in the **factor-size geometry** of:

$$
t\in[1/3,1/2]
$$

current Type-II / convolution technology overlaps strongly with this problem.

---

# 20. But the Square Cell Adds a Localization Gate

The core condition of SRBRD Goldbach-type convolution is an additive / congruence relation.

Legendre square-cell semiprimes, however, require:

$$
\boxed{
X<pr<X+H,
\qquad
H=X^{1/2}.
}
$$

This is an extremely short hyperbolic strip of relative thickness:

$$
\boxed{
H/X
=
X^{-1/2}.
}
$$

as specified above.

An ordinary dyadic:

$$
pr\asymp X
$$

convolution theorem does not automatically preserve this pointwise localization.

If a smooth weight replaces the indicator:

$$
\mathbf1_{(X,X+H]},
$$

its derivatives / Fourier scale introduce the localization cost:

$$
X/H
=
X^{1/2}
$$

of this scale.

Thus:

$$
\boxed{
\text{AP uniformity}
\neq
\text{short-product-window uniformity}.
}
$$

---

# 21. Public Provider Audit

## Maynard III

Provides prime AP mean-value theorems for moduli beyond $X^{1/2}$, with residue-class uniformity as its main feature.

This helps with:

$$
\boxed{
\text{residue gate}.
}
$$

But it does not automatically prove a pointwise prime-product upper bound in:

$$
X<pr<X+\sqrt X
$$

this short strip.

## Pascadi 2025

For primes / smooth numbers with triply well-factorable weights, the distribution exponent reaches:

$$
X^{5/8-o(1)}
$$

as specified above.

This improves:

$$
\boxed{
\text{modulus / structured-weight gate}.
}
$$

but is still not a square-anchor short-hyperbolic localization theorem.

## Yang 2026

Establishes a convolution-type Bombieri–Vinogradov theorem with well-factorable weights.

This is one of the public architectures closest to Provider B.

But it still requires an additional audit of:

$$
\boxed{
\text{sharp product localization to }H=X^{1/2}.
}
$$

---

# 22. The Meaning of Almost-All $E_2$ Results

Matomäki has proved:

Almost all:

$$
(x-h\log X,x]
$$

as:

$$
h\to\infty
$$

contain $P_2$.

Matomäki–Teräväinen also pushed exact $E_2$ to almost all polylogarithmic-length intervals.

Thus:

$$
\boxed{
\text{very-short bilinear }E_2\text{ technology exists}.
}
$$

But:

$$
\boxed{
\text{almost all anchors}
\neq
\text{every square anchor}.
}
$$

These results therefore cannot directly complete Provider B, still less Legendre.

---

# 23. Gafni–Tao Short-Interval Context

As of now, the known exponent for PNT in short intervals for **all** $x$ remains above:

$$
1/2.
$$

Recent work by Gafni–Tao on exceptional intervals points out that, using the latest zero-density estimates:

- All-$x$ PNT is known for $\theta>17/30$;
- The almost-all threshold can be much lower.

Our:

$$
H=X^{1/2}
$$

still lies just below the pointwise PNT threshold.

The square-cell pointwise constant ledger thus lies at the current short-interval analytic frontier.

---

# 24. Band Ledger

Phase 13 uses:

### Band A

$$
\frac13\le t\le\frac25,
$$

Ideal:

$$
\boxed{
C_A^*=\log\frac43.
}
$$

### Band H

$$
\frac25\le t\le\frac12,
$$

Ideal:

$$
\boxed{
C_H^*=\log\frac32.
}
$$

SRBRD-20 shows:

- Band A is the range most sensitive to Maynard-II fixed-residue / uniformization;
- Near $t\approx2/5$, there is a Maynard-III bridge;
- The middle-high range has a uniform Type-II architecture;
- The balanced edge has a restricted-uniformity architecture.

But this Legendre problem adds, across **all bands**, the common requirement:

$$
\boxed{
H=X^{1/2}
\text{ product localization}.
}
$$

---

# 25. Current Constant Ledger

The table that can honestly be filled in now is not:

| Provider | Constant |
|---|---:|
| Rough lower | $1+\log2$ |
| Semiprime upper | $\log2$ |

because that is only an ideal asymptotic.

The actual theorem-level ledger:

| Module | Desired constant | Current black-box theorem status |
|---|---:|---|
| Square-anchored cube-root rough lower | $C_R^->0$ | **OPEN at required pointwise scale** |
| SP low band upper | near $\log(4/3)$ | convolution shape available; **short-window localization OPEN** |
| SP high band upper | near $\log(3/2)$ | stronger uniform convolution available; **short-window localization OPEN** |
| Constant crossing | $C_R^- - U_{SP}>0$ | **NOT PROVED** |

Thus:

$$
\boxed{
\mathrm{PHASE13\ CURRENT\ BLACKBOX\ CROSSING=FAIL}.
}
$$

---

# 26. But the Main-Term Ledger Is PASS

The algebraic / heuristic main terms:

$$
\boxed{
(1+\log2)
-
\log\frac43
-
\log\frac32
=
1.
}
$$

Thus:

$$
\boxed{
\mathrm{MAIN\ TERM\ CONSTANT\ LEDGER=PASS}.
}
$$

The failure comes from provider strength, not the main-term constants of the architecture.

---

# 27. Finite Constant Audit

This round's machine audit:

$$
100\le n\le5000.
$$

Normalize by:

$$
\frac{\log X}{H}.
$$

Selected / block-averaged diagnostics are as follows:

| $n$ block start | avg prime coeff | avg low-SP coeff | avg high-SP coeff | avg rough coeff |
|---:|---:|---:|---:|---:|
| 100 | 0.999601 | 0.252560 | 0.372447 | 1.624608 |
| 600 | 0.999536 | 0.263867 | 0.386541 | 1.649943 |
| 1100 | 0.999579 | 0.266532 | 0.389826 | 1.655937 |
| 1600 | 0.999699 | 0.272625 | 0.393978 | 1.666302 |
| 2100 | 1.000528 | 0.275382 | 0.392526 | 1.668436 |
| 2600 | 1.000066 | 0.269289 | 0.397740 | 1.667096 |
| 3100 | 1.000157 | 0.280463 | 0.394574 | 1.675195 |
| 3600 | 0.999593 | 0.276875 | 0.396038 | 1.672506 |
| 4100 | 0.999992 | 0.275574 | 0.398047 | 1.673613 |
| 4600 | 1.000218 | 0.272544 | 0.397061 | 1.669824 |

The ideal constants:

$$
C_P^*=1,
$$

$$
C_A^*=\log\frac43\approx0.287682,
$$

$$
C_H^*=\log\frac32\approx0.405465,
$$

$$
C_R^*=1+\log2\approx1.693147.
$$

Finite block averages have stabilized around these scales.

This is only a diagnostic, not a uniform theorem.

---

# 28. A More Direct Parity-Signed Identity

Cube-root survivors consist only of:

- Primes: $\Omega=1$;
- Semiprimes: $\Omega=2$.

Let the Liouville function be:

$$
\lambda(q)=(-1)^{\Omega(q)}.
$$

Thus, for:

$$
q\in\mathcal S_3,
$$

we have:

$$
\lambda(q)
=
\begin{cases}
-1,&q\text{ prime},\\
+1,&q\text{ semiprime}.
\end{cases}
$$

Define:

$$
\boxed{
L_3(n)
=
\sum_{q\in\mathcal S_3(n)}
\lambda(q).
}
$$

Then:

$$
\boxed{
L_3(n)
=
E_2(n)-P_\square(n).
}
$$

And:

$$
S_3=P_\square+E_2.
$$

Solving gives:

$$
\boxed{
P_\square(n)
=
\frac{
S_3(n)-L_3(n)
}{2}.
}
$$

---

# 29. Ideal Parity Coefficient

From:

$$
P^*=1,
$$

$$
E_2^*=\log2,
$$

we obtain:

$$
\boxed{
C_L^*
=
\log2-1
=
-(1-\log2).
}
$$

That is:

$$
\boxed{
C_L^*
\approx
-0.30685281944.
}
$$

Thus the ideal cube-root rough population itself has a negative Liouville bias.

---

# 30. This Exposes the Parity Barrier Exactly

The standard sieve can study:

$$
S_3(n)
$$

and its unsigned rough mass.

But to cross from:

$$
P_2
$$

to:

$$
P_1,
$$

we must control:

$$
\boxed{
L_3(n).
}
$$

Thus:

$$
\boxed{
\text{parity barrier}
=
\text{lack of pointwise control of the signed rough Liouville mass}.
}
$$

This is more precise than ‘the sieve cannot distinguish primes / semiprimes’.

---

# 31. Signed Route

Besides the two-provider ledger:

$$
S_3-E_2,
$$

we can also take the route:

$$
\boxed{
P_\square
=
\frac{S_3-L_3}{2}.
}
$$

If we have:

$$
S_3(n)
\ge
C_R^-\frac H{\log X},
$$

and:

$$
L_3(n)
\le
C_L^+\frac H{\log X},
$$

then as long as:

$$
\boxed{
C_L^+<C_R^-,
}
$$

we obtain a prime.

The ideal gap:

$$
C_R^*-C_L^*
=
2.
$$

But current pointwise signed providers are even more lacking, so this is a:

$$
\boxed{
\text{parity-explicit strong route},
}
$$

not an easy route at present.

---

# 32. Ranking the Minimal Gaps of Phase 13

The present priorities can now be ranked explicitly.

## Gap 1 — PRIMARY

$$
\boxed{
\text{Square-Anchored Cube-Root Rough Lower}
}
$$

Requires:

$$
C_R^->0.
$$

The standard lower sieve is zero at $s=3/2$.

## Gap 2 — SECONDARY

$$
\boxed{
\text{Pointwise Short Hyperbolic }P\times P\text{ Upper}
}
$$

Requires sufficiently sharp bandwise constants.

## Gap 3 — PARITY DIRECT

$$
\boxed{
\text{Signed RootRough–Liouville Control}.
}
$$

This may directly cross the prime/semiprime parity boundary, but is currently the deepest route.

---

# 33. Phase 14 Should Stop Chasing Semiprime Constants

By the threshold audit:

If Provider A approaches the ideal value, Provider B has ample constant slack.

Phase 14 should therefore not first spend extensive resources changing:

$$
R_A,R_H
$$

from, for example, $1.5$ to $1.2$.

The question to answer first is:

> Is the square anchor sufficient for cube-root rough survivors to have a positive lower bound in every cell, preferably close to the Buchstab main term?

---

# 34. Phase 14 Target

## Square-Anchored Cube-Root Rough Lower

Study:

$$
\boxed{
S_3(n)
=
\sum_{1\le k\le2n}
\mathbf1_{
P^-(n^2+k)>n^{2/3+o(1)}
}.
}
$$

We need to exploit:

$$
\boxed{
k\equiv-n^2\pmod p
}
$$

rather than viewing it only as generic interval sieving.

Candidate tools:

1. Exact square-phase inclusion–exclusion;
2. Möbius-weighted discrepancy;
3. Quadratic phase averaging across divisor scales;
4. Shifted sieves;
5. Selberg / beta-sieve weights adapted to anchored residue phases;
6. A square-anchor spectral / Poisson transform;
7. Comparison with rough-number variance technology;
8. Finite exceptional-$n$ verification only after an analytic large-$n$ threshold is obtained.

---

# 35. Phase 13 Verdict

```text
Legendre Base-Space Program
Phase 13 — Square-Anchored P2 Constant Ledger

Exact P2 ledger:                            CLOSED
Ideal prime constant:                       1
Ideal semiprime constant:                   log 2
Ideal rough constant:                       1 + log 2
Low-band coefficient:                       log(4/3)
High-band coefficient:                      log(3/2)
Main-term crossing margin:                  +1
Band multiplier crossing region:            DERIVED
Finite constant diagnostics:                CONSISTENT
Standard linear rough lower at s=3/2:       ZERO / FAIL
Generic Jacobsthal route:                   TOO WEAK
Campbell P3 -> raw rough lower:              DOES NOT TRANSFER
Maynard/Pascadi/Yang factor geometry:        RELEVANT
Short hyperbolic localization gate:         OPEN
Current black-box constant crossing:         NOT PROVED
Parity-signed Liouville identity:            DERIVED
Primary missing provider:                   SQUARE-ANCHORED ROUGH LOWER
Legendre proof status:                      OPEN
Next phase:                                 SQUARE-ANCHORED CUBE-ROOT ROUGH LOWER
```

---

# 36. References

1. James Maynard, *Primes in arithmetic progressions to large moduli II: Well-factorable estimates*, Memoirs AMS **306** (2025), no. 1543; arXiv:2006.07088.
2. James Maynard, *Primes in arithmetic progressions to large moduli III: Uniform residue classes*, Memoirs AMS **306** (2025), no. 1544; arXiv:2006.08250.
3. Alexandru Pascadi, *On the exponents of distribution of primes and smooth numbers*, arXiv:2505.00653, 2025.
4. Zhiyuan Yang, *Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications*, arXiv:2608.13299, 2026.
5. Peter J. Campbell, *On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares*, arXiv:2603.10356, 2026.
6. Kaisa Matomäki, *Almost primes in almost all very short intervals*, J. London Math. Soc. **106** (2022), 1061–1097.
7. Kaisa Matomäki, Joni Teräväinen, *Almost primes in almost all short intervals II*, Trans. AMS **376** (2023), 5409–5432.
8. Ayla Gafni, Terence Tao, *On the number of exceptional intervals to the prime number theorem in short intervals*, arXiv:2505.24017, 2025.
9. Henryk Iwaniec, *On the problem of Jacobsthal*, Demonstratio Mathematica **11** (1978), 225–232.
10. EveMissLab internal SRBRD-13–20, 2026-09-30.